central difference approximation
Where can I find the expression of the central difference approximation of the first and second derrivative (spatial) for a NON uniform grid?
For 3 points, (x1, f1), (x2, f2), (x3, f3), x1 < x2 <x3
Two methods (identical results for the approximation):
1. Expand f1 and f3 about x2 in Taylor series. Expand through the second derivatives.
You get two equations with unknowns df/dx and d^2f/dx^2 evaluated at x2. Solve those for your solution. Note: Expand through the 3rd and 4th derivatives, you have terms for the major errors you ignore by truncating the series after the 2nd derivatives.
2. Fit a parabola, f(x) = f2 + b(x - x2) + c(x - x2)^2 through the 3 points. b and c will be functions of x1, x2, x3 and f2 and f3.
Try this book:
Ferziger and Peric - Computational Methods for Fluid Dynamics
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